A ring of mass m and radius r rolls without slipping on a fixed hemispherical surface of radius R as shown. The time period of small oscillations of ring is `2pisqrt((n(R-r))/(3g))` then find the value of n.

Answer

6

Explanation

`mv^(2)+mg(R-r)(1-costheta)=` constant
`implies2mv(dv)/(dt)+mg(R-r)sintheta(dtheta)/(dt)=0`
since `v=omega(R-r)=(-dtheta)/(dt)(R-r)`
`(dv)/(dt)=(-d^(2)theta)/(dt^(2))(R-r)`
`2m(dtheta)/(dt)(R-r)(d^(2)theta)/(dt^(2))(R-r)+mg(R-r)sintheta(dtheta)/(dt)=0`
`implies(d^(2)theta)/(dt^(2))=(-gtheta)/(2(R-r))` (for small `theta,sintheta=0`)
`impliesT=2pisqrt((2(R-r))/(g))`